Saddlepoint Approximation for Linear Function of Uniformly-Distributed Data

We follow the method of Section 2.3.2. Let ${\bf x}$ be a set of $N$ independent uniform-distributed RV in $[0,1]$ Let ${\bf A}$ be an $N$-by-$D$ full-rank matrix and let ${\bf z}$ be the $D\times 1$ feature vector

$${\bf z}={\bf A}^\prime {\bf x}.$$

When ${\bf x}\in{\cal R}^N$ is uniformly distributed, the CGF is given by

$$c_z($$$\lambda$$$)=\sum_{n=1}^N \log \left( \frac{\exp\left( \sum_{i=1}^D \lambda_i A_{n,i}\right)-1}{\sum_{i=1}^D \lambda_i A_{n,i} }\right).$$

For conciseness, define

$$w_n=\sum_{i=1}^D \lambda_i A_{n,i}.$$

Then,

$$c_z($$$\lambda$$$)=\sum_{n=1}^N \log \left( \frac{e^{w_n}-1}{w_n}\right),$$

$$\frac{\partial}{\partial \lambda_i} c_z($$$\lambda$$$)=\sum_{n=1}^N \left( \frac{e^{w_n}}{e^{w_n}-1} - \frac{1}{w_n}\right) A_{n,i},$$

and

$$\frac{\partial^2}{\partial \lambda_i \partial \lambda_j} c_z($$$\lambda$$$)=\sum_{n=1}^N \left( \frac{e^{w_n}}{e^{w_n}-1} - \frac{(e^{w_n})^2}{(e^{w_n}-1)^2} + \frac{1}{w_n^2}\right) A_{n,i} A_{n,j}.$$

From these expressions, we may find the saddle-point (2.12).